2004/03/10 by Bernd Ammann, Chad Sprouse, C. Sprouse · 1 citation
Mathematics · #Advanced Operator Algebra Research #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #math.SP #msc:53C20 #msc:53C21. #msc:53C27 #msc:58J50
paper · pdf · doi:10.1007/s10455-006-9048-2
published as Ann. Glob. Anal. Geom. 31, 409-425 (2007)
arxiv created 2004/03/10 · openalex publication_date 2006/11/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Consider the class of n-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let r=1 if n=2,3 and r=2[n/2]-1+1 if n≥ 4. We show that if the square of the Dirac operator on such a manifold has r small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. Equivalently, if M is not a nilmanifold or if M is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the r-th eigenvalue of the square of the Dirac operator. If a manifold with almost nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on M with a parallel spinor. In dimension 4 this implies that M is either a torus or a K3-surface.